Miraculous cancellations for quantum $SL_2$

TL;DR

This paper explores special cancellations in quantum trace maps at roots of unity, providing a representation theoretic interpretation involving quantum groups and their duals.

Contribution

It offers a new understanding of miraculous cancellations in quantum trace maps through the lens of quantum group representation theory.

Findings

Representation theoretic interpretation of cancellations

Connection between quantum trace maps and quantum groups

Insights into quantum $SL_2$ at roots of unity

Abstract

In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter $q$ is a root of unity. The current paper is devoted to giving a more representation theoretic interpretation of this phenomenon, in terms of the quantum group $U_q(sl_2)$ and its dual Hopf algebra $SL_2^q$.